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\documentclass[aip,graphicx]{revtex4-1}
\usepackage{axodraw}
\usepackage{color}
\usepackage{graphicx}
\usepackage{float}
\usepackage{amsmath}
%\documentclass[aip,reprint]{revtex4-1}

\draft % marks overfull lines with a black rule on the right

\begin{document}

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%\preprint{}

\title{Bruckener-Hartree-Fock} %Title of paper

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\author{Zhonghao Sun}
\email[]{chonghawk.sun@gmail.com}
%\homepage[]{Your web page}
%\thanks{}
%\altaffiliation{}
\affiliation{Peking University}

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\date{\today}

\begin{abstract}
basic formulars of bhf
\end{abstract}

%\pacs{}% insert suggested PACS numbers in braces on next line

\maketitle %\maketitle must follow title, authors, abstract and \pacs

% Body of paper goes here. Use proper sectioning commands. 
% References should be done using the \cite, \ref, and \label commands
\section{HF framwork}
%\label{}
In HF basis, sigle particle energy was influnced by the core, Fig.~\ref{fig:hf} shows the physics behind, HF matrix is:
     \begin{equation}
      h_{ac}=T_{ac}+\sum_h<ah|v|ch>\rho_h
     \end{equation} 
     For simplification, We omite knetic energy in following equations.
     
     \begin{center}
     \begin{figure}[H]
\includegraphics[width=0.4\textwidth]{img/hf.pdf}
\caption{\label{fig:hf}}
\end{figure}
 \end{center}
 
In ho basis:
    \begin{equation}
     \sum_{ij}D_{ai}D_{cj}h_{ij}=\sum_h\sum_{kl}\sum_{ij}D_{ai}D_{cj}D_{hk}D_{hl}<ik|V|jl>
    \end{equation}
    \begin{equation}
     \sum_{ij}D_{ai}D_{cj}h_{ij}=\sum_{ij}D_{ai}D_{cj}<ik|V|jl>\rho_{kl}
    \end{equation}
    with $\rho_{kl}=\sum_h D_{hk}D_{hl}$
   \begin{equation}
     \sum_{ij} \sum_{ac}D^\dag_{ai'}D^\dag_{cj'}*D_{ai}D_{cj}h_{ij}=\sum_{ac}D^\dag_{ai'}D^\dag_{cj'}*\sum_h\sum_{kl}\sum_{ij}D_{ai}D_{cj}D_{hk}D_{hl}<ik|V|jl>
    \end{equation}
  \begin{equation}
     \sum_{ij} \delta_{ii'}\delta_{jj'} h_{ij}=\sum_{ij}\delta_{ii'}\delta_{jj'}\sum_{kl}<ik|V|jl>\rho_{kl}
    \end{equation}

     \begin{equation}
     h_{ij}=\sum_{kl}<ik|v|jl>\rho_{kl}
    \end{equation}
    
     \begin{equation}
     h_{ij}=t_{i,j}+\sum_{kl}<ik|v|jl>\rho_{kl}
    \end{equation}
\section{1st HF}

     \begin{center}
     \begin{figure}[H]
\includegraphics[width=0.6\textwidth]{img/1st.pdf}
\caption{\label{fig:1st}1st correction to hf}
\end{figure}
 \end{center}
 The left diagram contribute:
 
    \begin{equation}
     1st_a)=\frac{1}{4\hat{j_a}^2}\sum_{JT}\hat{J}^2\hat{T}^2\sum_{p1 p2}\frac{<ah;JT|V|p_1p_2;JT><p_1p_2;JT|ch;JT>}{\epsilon_c+\epsilon_h-\epsilon_{p_1}-\epsilon_{p_2}}
    \end{equation}
 
 
 
     \begin{center}
     \begin{figure}[H]
\includegraphics[width=\textwidth]{img/obir.pdf}
\caption{\label{fig:obir}}
\end{figure}
 \end{center}
 
 In hf basis:
     \begin{equation}
     \begin{split}
      & \sum_{p_1 p_2 h}<ah|v|p_1p_2><p_1p_2|v|ch> \\
      =& \sum_{p_1 p_2 h} \sum_{ij} \sum_{pq} \sum_{st} \sum_{mn}D_{ai}D_{cj}D_{hq}D_{p_1s}D_{p_2t}D_{p_1m}D_{p_2n}D_{np} \\
      & <iq|v|st>^{JT}<mn|v|jp>  
     \end{split}
     \end{equation}

   \begin{equation}
    <a|\chi|c>=\sum_{pq}\sum_{st}\sum_{mn}\frac1{4\hat{j_a}^2}\sum_{JT}\frac{<iq|v|st>^{JM}<mn|v|jp>^{JM}}{\epsilon_c+\epsilon_h-\epsilon_{p_1}-\epsilon_{p_2}}
    \rho_{pq}^{hole}\rho_{sm}^{part}\rho_{tn}^{part}
   \end{equation}
      
      with:
      \begin{equation}
       \rho_{ij}^{hole}=\sum_{h=1}^AD_{ih}D_{jh}
      \end{equation}
      \begin{equation}
       \rho_{ij}^{part}=\sum_{h=A+1}^{NMAX}D_{ih}D_{jh}
      \end{equation}

$\rho^{hole}$ is the probablity of the orbit being occupied, and $\rho_{part}$ is that of not being occupied. this is equal to
the pauli exlusion in nuclear matter calculation.

All diagram contribute to Hartree forck is:
  \begin{center}
     \begin{figure}[H]
\includegraphics[width=\textwidth]{img/2st.pdf}
\caption{\label{fig:obir}}
\end{figure}
 \end{center}
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